Fixed Rank co-Kriging: a model for multivariate spatial prediction

2026-03-23

AI Summary

This work proposes a multiresolution cokriging model to address the challenge of sparse or missing observations in multivariate spatial prediction. By jointly modeling latent effects across multiple correlated spatial processes, the method enables information sharing and collaborative prediction. It integrates spatial basis functions with Gaussian Markov random field coefficients, explicitly capturing cross-process dependence without assuming global stationarity, while preserving the computational efficiency of fixed-rank kriging. Parameter estimation is carried out via an expectation–maximization algorithm. Simulation studies demonstrate that the approach effectively leverages densely observed variables to improve prediction accuracy in regions with sparse or no observations. The practical utility of the method is further validated through an application to PM10 concentration prediction in northern Italy.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This work develops a multivariate extension of the Fixed Rank Kriging (FRK) framework for spatial prediction in settings where multiple spatial processes may provide complementary information. The goal is to preserve the computational efficiency, the ability to operate without assuming stationarity over the domain, and the spatial support flexibility of FRK, while incorporating cross-process dependence. To this end, we employ a multiresolution coregionalization structure for the latent spatial effects, in which spatial basis functions are combined with Gaussian Markov Random Field coefficients. An estimation procedure based on the expectation-maximization algorithm is developed, designed to exploit the multiresolution latent structure. Through simulation studies, we examine when the proposed joint modeling is beneficial. We consider cases in which one process is observed more sparsely or is entirely unobserved in a subregion and find that the multivariate formulation is able to borrow information from the more densely observed process, producing coherent and accurate predictions even where direct observations are limited or absent. Finally, the model is applied to the analysis of PM10 concentrations in Northern Italy, illustrating its applicability in a real environmental context.

Problem

• multivariate spatial prediction
• cross-process dependence
• spatial kriging
• missing observations
• environmental monitoring

Innovation

• Fixed Rank co-Kriging
• multiresolution coregionalization
• Gaussian Markov Random Field
• expectation-maximization algorithm
• multivariate spatial prediction